What is a good way to make large stylized caverns/caves?
A possibility could be to use the cell fracture add-on for Blender.

Basically, the cell fracture tool can destruct an object into multiple ones,  fracturing it into lots of pieces. 

My idea: use it on a cube, and make the pieces fairly small. Run the simulation for a few frames, then delete all the chunks you wanted, leaving a nice cavernous interior.  After that, use the Randomize Transform tool to add some more variation. 

Once that's finished, all that's left is textures, materials, and lighting! 

Good luck. ;-P